1. What is Trigonometric Form of Complex Numbers?

The trigonometric form (also called polar form) represents a complex number using its magnitude and angle. Any complex number z = a + bi can be written as z = r (cos θ + i sin θ), where r is the magnitude and θ is the angle.

2. How Does the Calculator Work?

The calculator uses the following formulas:

Where:

• \( a \) — Real part of the complex number
• \( b \) — Imaginary part of the complex number
• \( r \) — Magnitude (always non-negative)
• \( \theta \) — Angle in radians (between -π and π)
• \( i \) — Imaginary unit (√-1)

Explanation: The trigonometric form is particularly useful for multiplication and division of complex numbers, as well as for raising complex numbers to powers.

3. Importance of Trigonometric Form

Details: The trigonometric form simplifies complex number operations and is essential in fields like electrical engineering, signal processing, and quantum mechanics.

4. Using the Calculator

Tips: Enter the real and imaginary parts of your complex number. The calculator will show the trigonometric form and detailed solution steps.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between rectangular and trigonometric forms?
A: Rectangular form is z = a + bi, while trigonometric form is z = r(cosθ + i sinθ). Trigonometric form shows magnitude and angle.

Q2: How do I convert back to rectangular form?
A: Use a = r cosθ and b = r sinθ to get back to a + bi form.

Q3: What range does θ have?
A: The angle θ is between -π and π radians (-180° to 180°).

Q4: Can r be negative?
A: No, r (magnitude) is always non-negative.

Q5: Why use trigonometric form?
A: It simplifies multiplication (multiply magnitudes, add angles) and exponentiation (De Moivre's Theorem).